A sum of ₹ 18750 is left by a will by a father to be divided between the two sons, 12 and 14 years of age, so that when they attain maturity at 18, the amount (principal + interest) received by each at 5 per cent simple interest will be the same. Find the sum allotted at present to each son. (N.M.A.T., 2005)

### Concept & Equating Amounts When a sum is divided into parts such that their final amounts (Principal + Interest) are equal, the ratio of the principals is inversely proportional to $(100 + RT)$ for each part. $$A = P \times \left(1 + \frac{R \times T}{100}\right)$$ ### Step-by-Step Solution * **Given:** Total Sum = ₹ 18,750. Rate ($R$) = 5% p.a. * Younger son's age = 12 years. Time ($T_1$) to reach 18 = $18 - 12 = 6$ years. * Older son's age = 14 years. Time ($T_2$) to reach 18 = $18 - 14 = 4$ years. * **Calculation:** Let the younger son's share be $x$. The older son's share is $(18750 - x)$. * Amount for younger son = $x + \frac{x \times 5 \times 6}{100} = x + 0.30x = 1.30x$ * Amount for older son = $(18750 - x) + \frac{(18750 - x) \times 5 \times 4}{100} = (18750 - x) \times 1.20$ * Equate the amounts: * $1.30x = 1.20(18750 - x)$ * $1.3x = 22500 - 1.2x$ * $2.5x = 22500$ * $x = \frac{22500}{2.5} = 9000$ * Younger son's share = ₹ 9,000. * Older son's share = $18750 - 9000$ = ₹ 9,750. ### Exam Strategy & Shortcut Use the direct ratio formula for equal amounts: $P_1 : P_2 = (100 + R_2 T_2) : (100 + R_1 T_1)$. $P_1 : P_2 = (100 + 5 \times 4) : (100 + 5 \times 6) = 120 : 130 = 12 : 13$. Younger son's share ($P_1$) = $\frac{12}{25} \times 18750 = 9000$. Older son's share = $\frac{13}{25} \times 18750 = 9750$. ### Common Pitfall Students often mistakenly equate the simple interests instead of the total amounts, ignoring the "+ principal" part of the condition. ### Final Answer Therefore, the correct answer is **₹ 9000, ₹ 9750**.